If you are referring to the aikijo and aikiken of Saito M, which is what Kobayashi Y. adopted as well, then this is what it is about:

Ken awase and Jo awase refer to sets of exercises to practice the basic movements (suburi etc) with a partner and they add [to the solo forms] the properties of awase (blending), maai (distancing/spacing), connection, timing and control of attack-line.

That is, these sets are NOT meant to be "applied" techniques. Instead they represent an intermediary step between solo training and kumi-practice (combative forms of paired practice).

They do NOT have to be mirrored. What is important is that uchi tachi respectively uchi jo (attacker) gives uke tachi or uke jo (defender) various forms of attacks to practice the above mentioned properties.

Now, Saito initially did a set of 8 jo awase, found in Saito's original books, and one of which was shown by Inocencio above. They have similar purpose as the present ken awase (right, left awase + 5 and 7 awase) but Saito M. later stopped teaching these. I feel these awase are great tools, and my guess is that Saito M. wanted to keep the number of partner practices down.

He also had the 2 jo awase forms where he used the two major kata as template: 13 kata and 31 kata. In both these cases uchi jo provides attacks and uke jo blend and defend each attack movement (e.g. 1-3, 3-6 etc. in 31 kata) by "winning". In this way uke tachi is able to practice many different awase forms and defenses. One attack - one defense.
Lately 31 jo awase has been taught only as the 31 kumi jo, where the stops have been taken away and it doesn't finish until the end of the full kata. 13 jo awase however has been kept intact.

In all of the above cases the awase and kumi jo represent ways for Saito to teach us his lesson from O-sensei. O-sensei did not teach these partner practices. He taught Saito M. the solo 31 kata and Saito sensei used 13 kata as a way to represent the rest of the solo Jo work that was missing in 31 kata he was shown by the founder. What the founder DID do according to, Saito M, was to teach simple applications of the jo:
- He attacks like this - You can defend like this or like that.... etc.
Simple and short. Probably not unlike the first set of 8 jo awase.
That's it! Nothing else. The rest, again, was Saito's way of teaching what he found was the lesson, using the means he had:
Suburi - 31 Kata and 13 kata.

Now, going into the kumi tachi/jo set. They add [to the suburi and awase] the dimension of application and realism: Here you have to learn and train to move outside the box of perfect basics into the world of adrenaline, dynamics and non-linearity of combat! BUT... you do this as an application of AIKI.
Particularly in the case of kumi tachi the Founder used old-schooled forms (mostly from Kashima Ryu) as a template and then reshaped them for this expression of high level aiki with sword. Saito M. kept the Founders forms in their entirety.
In the case of kumi jo - some were the Founders creation and others were Saito Sensei's own developments based on the founders teachings.

Finally wrt. Kobayashi Y. Sensei: His student Igarashi sensei went to Iwama to train with Saito sensei and brought back the buki waza, which Kobayashi since that point implemented into his curricula. What Kobayashi sensei added was how to practice some of the awase forms. For example the way in which he let both mirror each other in 13 awase. In contrast to the way Saito taught it. He also kept the 8 jo awase forms.